When we teach mathematics in grade seven to nine we quite often meet the misconception “multiplication always increases a number and division always decreases a number”. This was the starting point for our learning study conducted in Sweden during four months in 2011. In this learning study we wanted our 74, 8ͭ ͪ grade students, to enhance their understanding of division and we also wanted to enhance our teaching. In more specific terms and in terms of the object of learning; we wanted the students to notice why the result in division sometimes becomes greater than the numerator.

In order to investigate student understanding, a pre-test and a post-test were used before and after the lessons. The tests were analyzed with the intention to identify critical aspects related to the object of learning. The students’ different understandings were a useful point of departure when the lessons were planned. The study also consisted of four video-recorded lessons. In this study, variation theory (Marton & Tsui, 2004) was used as a framework to analyze the relation between what is taught and what is learned.

In this presentation we will give examples from the differences between two of the lessons, A and B, and how we used variation theory to handle the critical aspects in different ways. It was found from the pre-test that some students perceived the two tasks 100 • 0,5 and  100/0,5 as equal. It was also found that several students just saw the sharing aspect of division and not the measuring aspect. Those findings were clues to understand what was critical and what was not. The results from the lessons showed that different dimensions of variation in the task design, contributed to differences in student learning outcomes. In lesson A, some of the critical aspects were just handled. This seemed to make the students notice calculation strategies. In lesson B, however, the critical aspects were handled simultaneously in the taskdesign, which affected the students’ learning in a different and more powerful way.